The Battle of New Orleans was fought in 1815, during the War of 1812.  The battle actually was the last battle of the conflict, and was fought after the Treaty of Ghent officially ended the War.  Another clue in this question is that Andrew Jackson was the General in command, where the Mexican-American War was fought after he was President, and the other answer choices are Wars fought before Jackson even served in the Army.

At the time of the American Revolution, Mexico was not yet an independent nation, but was still under the direct control of Spain.  Many Europeans came to America to support the colonists' cause, however, including the Marquis de Lafayette of France, Theodore Kosciuszko and Casimir Pulaski of Poland, and Bernardo de Galvez of Spain.  Germans fought on both sides of the American Revolution in different capacities.

New York was founded by the Dutch in 1614 as New Amsterdam, and fully settled by the Kingdom of the Netherlands in 1621.  All of the other answer choices were colonies founded by the English.  The best clue to the correct answer is the name Pieter Stuyvesant, which should indicate the correct answer will be a non-English settlement.

Spain and France were regional and global rivals to the British Empire at the time of the Revolutionary War. France, in particular, required little convincing to provide financial support to the colonial revolution. As the war progressed and it became clearer that the Americans could, and probably would, emerge victorious, France began to contribute more and more to the Revolutionary cause. They sent out military tacticians and generals to assist in training, discipline and strategy. They provided arms and ships. And eventually joined the war effort and helped end the war with the assistance of their navy. 

The Consitution enumerates multiple powers given explictly to Congress in Article One, that are not given to the other branches of the government. Among these is the specific power to declare war on behalf of the United States Government. Despite this, the President is Commander in Chief of the United States military, and is in charge of the military in any conflict.

In Paris, on February 6th, 1778, the United States of America signed a treaty of alliance with France, promising military support for the U.S., in the face of British attacks. It is known as the Treaty of Allliance with France.

In 1583, Queen Elizabeth I of England decided to begin colonizing North America, particularly because rivals France and Spain had already established settlements of their own. Sir Walter Raleigh, one of Elizabeth's favorites and noted landowner in Ireland, was given charge of settling Virginia, then considered a broad swath of the middle of the continent, and founded a settlement called Roanoke. The colony was star-crossed, however, and when Raleigh returned to check on it in 1587, no trace of the colony could be found. Permanent, lasting English settlement was not achieved until 1603 at Jamestown, north of the Roanoke site.